Search

We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others. [Copyright &y& Elsevier]

Abstract: Based on the geometric grid information as geometric coordinates, an algebraic multigrid (AMG) method with the interpolation reproducing the rigid body modes (namely the kernel elements of semi-definite operator arising from linear elasticity) is constructed, and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient for such semi-definite problems, and the convergence is uniformly bounded away from one independent of the problem size. Furthermore, the AMG method proposed in this paper has better convergence rate than the commonly used AMG methods. Simultaneously, an AMG method that can preserve the quotient space, which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered, then the numerical solution of AMG method is convergent in the same quotient space, is obtained using the technique of orthogonal decomposition. [Copyright &y& Elsevier]